Checking semidefiniteness

Question

I have a question regarding the definition of semidefinite matrices. Let's say we have a twice differentiable function $f: \mathbf{R}^2_{++} \rightarrow \mathbf{R}$ and want to check if $f$ is convex. We derive the Hessian $\mathbf{H}$ of $f$ and check its semidefiniteness which requires us to show that $v^T\mathbf{H}v\geq 0$. My question is do we have to take $v$ to be in $\mathbf{R}^2_{++}$? or can it be any vector on $\mathbf{R}^2$?